If champagne is frozen - will it be drinkable when thawed?
Yes, although being a liquid it will expand and break the bottle
How long it will last will depend upon how good it was when frozen and how it was thawed. If it was thawed at room temperature, it should be cooked immediately. If it wasn't fresh when frozen, it should be cooked immediately. If it was very fresh when frozen and was thawed in the fridge, then you should have at least a couple days to cook it.
No, frozen is frozen. But you can refreeze something that has been thawed. You can refreeze thawed meat long as there are still ice crystals on it. Once completely thawed then you need to cook it first then refreeze. This applies to solid pieces only, ground meat once more than half thawed needs to be cooked first then refrozen always.
Can frozen hamburgers that thawed on ice in a cooler be re-frozen and then cooked and eaten without making you sick?
If it was handled and packaged properly, was safe before originally frozen, was thawed under refrigeration and re-frozen without allowing the thawed meat to become contaminiated or to go bad, it could be safe. It probably won't taste as good or might be drier and tougher than if it had not gone through an extra freeze-thaw cycle.